2.1.3 Study Project: Functions to a proper subset Study Project: Functions to a proper subset In this study, we consider functions from a set S to a proper subset of S , comparing the situations where S is finite and where S is denumerable. The following theorems will be helpful: Theorem on finite cardinalities: A nonempty finite set is in bijection with  k for a unique k ∈ . Composition of bijections: If α : A → B and β : B → C are bijections, then β  α : A → C is a bijection. 1. Functions to a proper subset of a finite set Let S = {1,2,3,..., 20} . Consider the subset S ' = {5, 10, 15, 20} of all multiples of 5 in S . Either give an example, or use the theorems above to explain why no such example exists: a) A function f : S → S ' that is surjective but not injective b) A function g : S → S ' that is neither surjective nor injective c) A function h : S → S ' that is both surjective and injective Hint: Show that there exist bijections b : S →  20 and b': S ' →  4 . Now suppose there also exists a bijection h : S → S ' . Using the two theorems above, what do we find? d) A function k : S → S ' that is injective but not surjective 2. Functions to a proper subset of a denumerable set Let S = {1,2,3,...} . Consider the subset S ' = {5, 10, 15, ...} of all multiples of 5 in S . Either give an example, or explain why no such example exists: a) A function f : S → S ' that is surjective but not injective b) A function g : S → S ' that is neither surjective nor injective c) A function h : S → S ' that is both surjective and injective d) A function k : S → S ' that is injective but not surjective Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810 2.1.3 Study Project: Functions to a proper subset 3. Generalizing the results Let S be a countable set (finite or denumerable), and let S ' be a nonempty proper subset of S . a) Explain how to construct a function f : S → S ' that is surjective but not injective, regardless of the countable cardinalities of S and S ' . b) Under what conditions on S and S ' does there exist a function f : S → S ' that is neither surjective nor injective? Hint: Can S ' contain only one element? What about S ? c) Does there exist a bijection h : S → S ' regardless of the countable cardinalities of S and S ' ? If not, what conditions on the cardinalities of S and S ' must be met? d) Give some examples of your own to show that, depending on the countable cardinalities of S and S ' , there may or may not exist a function k : S → S ' that is injective but not surjective. 4. A final check True or False? Prove your answer! Suppose S ' is a nonempty proper subset of S and φ : S → S ' is a function that is not a bijection. Then S and S ' do not have the same cardinality. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-­‐0837810